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We can epitomize the datuming algorithm in a depth
varying velocity by writing the successive operators
in matrix form. For simplicity I will take only
a datum with three depth levels
similar to the one in Figure .
The algorithm can be generalized for any number of
depth levels and any datum geometry.
For each value of we have
 
(6) 
where represents the zerooffset data after
Fourier transformation along the time axis.
The matrix
performs the Fourier Transform matrix of the horizontal space variable.
Therefore the matrix
is composed of inverse Fourier Transform block matrices, each being
the conjugate transpose to the direct transformation.
The matrix
contains three blocks corresponding to the
product of diagonal matrices of the form
 
(7) 
On each diagonal are the phaseshifting exponentials
necessary to upward extrapolate the wavefield to a depth level.
The value of (k_{z})_{ij} is given by the dispersion relation:
Datumatrix
Figure 2 Datuming for only three depth levels. Data extrapolated to the
first depth level is extracted by matrix A, to the second
level by matrix B and to the third by matrix C.

 
The matrices A,B,C are matrices corresponding
to the shape of the datum.
For the case shown in Figure the matrices A,B,C are
The conjugate transpose algorithm can be obtained by reversing
the order of matrix multiplication in equation (6)
and transposingconjugating each matrix. The matrices
A,B,C are diagonal matrices and therefore equal to the
transpose.
For each value of the frequency the conjugate datuming is
 
(8) 
where represents the wavefield recorded on
the topographic datum. The matrices W^{*}_{i} are conjugate transposed
of the diagonal matrices W_{i} which are explicitly represented in
equation (7).
Next: Waveequation Datuming for laterally
Up: WAVEEQUATION DATUMING ALGORITHM
Previous: WAVEEQUATION DATUMING ALGORITHM
Stanford Exploration Project
11/17/1997